30 problems

Let $x\in\mathcal{X}$ and let $V\in \operatorname{Irr}\,\mathrm{U}^\imath_{v,x}$. Let $\varphi(x)\in G$ be the point associated with $x$, and let … be the $\theta$-twisted conjugac…

Let $(Q,\tau)$ be a general $\imath$quiver, allowing loops, and let the quantum symmetric pair and Hall algebra constructions associated with $(Q,\tau)$ be defined as in Theorem. H…

Let $V$ be an irreducible representation of $\mathrm{U}^\imath_{v}$, and let $a(V)$ be the dimension of the $\theta$-twisted conjugacy class containing $\Psi(V)$. Here $\ell$ is th…

Let $(\mathbf{U},\mathbf{U}^{\imath})$ be an iquantum group of finite type, where $\mathbf{U}^{\imath}$ is the coideal subalgebra of a Drinfeld–Jimbo quantum group $\mathbf{U}$ ass…

Reduced-expression independence conjecture. The element $\mathfrak{X}_{\widetilde{w}}$ depends only on $\widetilde{w}\in\widetilde{W}$ and not on the chosen reduced expression. The…

Let $({\mathfrak g},{\mathfrak g}^\theta)$ be an irreducible symmetric pair, let ${\check{B}}$ be its quantum symmetric-pair coideal subalgebra, and let $\mu\in P_{Z({\check{B}})}$…

Let $\mathsf{X}_i^{(k)}$ denote the nonclassical divided-power web at position $i$ in $\mathbf{Web}(\mathfrak{so}_{2n+1})$, with $1\leq k\leq n$. Divided-power exchange relation co…

Let $(V,\mathbb{B}_V)$ be a based $U_{-q^2}$-module, for example a finite-dimensional irreducible type I representation with Lusztig canonical basis $\mathbb{B}_V$. Let…

Balagović–Kolb's fundamental lemma. The formula

AIII dual canonical basis positivity conjecture. The dual canonical basis of $\widetilde{{\mathbf U}}^\imath$ of type ${\rm AIII}_{2n}$ has the properties outlined in the positivit…

Kirillov–Reshetikhin K-matrix conjecture. Symmetrizable invertible solutions of the generalized spectral reflection equation in tensor products of Kirillov–Reshetikhin modules, wit…

Meromorphicity conjecture. For all $V\in\mathcal{C}$, not necessarily irreducible, and for generic quantum symmetric pair parameters, $K_V(z)$ is the Laurent series expansion of a…

Affine factorization conjecture. In affine type, $\Upsilon$ admits an infinite-product factorization analogous to such a factorization of the quasi-$R$-matrix of quantum affine alg…

Generalized-parameter conjecture. For every $(\gamma,\sigma)\in\Gamma\Sigma_q$, the pair $(U_q(\mathfrak{g}),\mathcal{B}_{\gamma,\sigma})$ admits a cylindrical structure. Moreover,…

Let $\cA_q$ be the alternating central extension of the q-Onsager algebra, and let $V$ be an evaluation representation of $U_q(\widehat{\mathfrak{sl}}_2)$. Consider the certain exp…

Let $i$ be a vertex of type (iii) in the Satake-diagram setting, let $\widetilde{\mathbf{U}}^\imath$ be the universal $\imath$quantum group, and let…

Let $(\mathbb{I}=\mathbb{I}_{\circ}\cup\mathbb{I}_{\bullet},\tau)$ be a Satake diagram of finite type, let $\mathbf{U}^\imath_{\boldsymbol{\varsigma}}$ be the associated $\imath$qu…

Let quasi $K$-matrices be the quantum symmetric pair analogues of quasi $R$-matrices, and let rank $1$ quasi $K$-matrices denote the corresponding rank-one factors. Dobson–Kolb fac…

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, let $\theta:\mathfrak{g}\to\mathfrak{g}$ be an involutive Lie algebra automorphism of the second kind, and let…

Parameter-inversion conjecture. There exists $g\in(U_q\mathfrak h)^{\mathcal W,\times}$ such that this relation is satisfied. Consequently, $k_{Y,\eta}$ specializes to a solution o…

Let $X \subseteq I$ be of finite type and let $\tau \in \operatorname{Aut}_X(A)$ satisfy $\tau^2=\operatorname{id}_I$ and $\tau|_X=\operatorname{id}_X$. For…

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, let $(X,\tau)$ determine a quantum symmetric pair coideal subalgebra…

Let $(I,X,\tau)$ be a Satake diagram of finite type, let $\widetilde{W}$ be its restricted Weyl group, and let $\widetilde{w_0}$ be the longest element. For a reduced expression of…

A quantum symmetric pair is specified by a Satake diagram and parameters, and its quasi $K$-matrix is the intertwiner $\mathfrak{X}$ between the corresponding bar involutions. The…

Let quantum symmetric pairs be quantum group analogues of symmetric pairs, and let the underlying Kac–Moody type be finite or, more generally, Kac–Moody. Braid group action conject…